# In memoriam of Stasys Rutkauskas (Lithuania) and Garnik Karapetyan (Armenia)

The texts below are from the volume "Modern Methods in Operator Theory and Harmonic Analysis" OTHA 2018, Rostov-on-Don, Russia, April 22-27, Selected, Revised and Extended Contributions. DOI:10.1007/978-3-030-26748-3 . Editors: Alexey Karapetyants, Vladislav Kravchenko and Elijah Liflyand.

**Prof. Garnik Karapetyan
1958/02/03-2018/11/29**

The diverse scientific activity of G.Karapetyan has made him well-known in the mathematical community. One of the main parts of the theory of differential equations and functional analysis, the general theory of differential equations in the spaces of differentiable functions, is associated with his name. His research has played a significant role in development of a general theory of integral representations and embedding theorems in multianisotropic functional spaces.

Garnik Karapetyan was born on February 3, 1958 in the city of Leninakan (now Gyumri), Armenian SSR. In 1975, he started undergraduate studies at the Faculty of Mechanics and Mathematics of Yerevan State University (YSU) and graduated cum laude in 1980. In 1983 he had early defense of PhD thesis and started to work as assistant at the Department of Numerical Analysis of the Faculty of Informatics and Applied Mathematics of YSU. From 1990 till 1994, he held dean position of the Faculty of Informatics and Applied Mathematics of YSU. Garnik Karapetyan was at the head of the Mathematics Olympiad Committee of Armenia for two years starting from 1997. In 2002, he founded the Department of Mathematics and Mathematical Modeling of Russian-Armenian University (RAU). In 2007, Garnik Karapetyan successfully defended the Doctor of Science thesis titled “Properties of the solutions of regular hypoelliptic equations”. In 2008, he received the academic rank of Professor from the Higher Attestation Commission of the Republic of Armenia and the title of Distinguished Professor of RAU. Starting from 2008, he was a member of the Scientific Council of RAU and Dissertation Council in Mathematics of the Republic of Armenia. From 2015, he is Director of the Applied Mathematics and Informatics program at RAU. He was the first Director of the Institute of Mathematics and Informatics of RAU. Garnik Karapetyan was awarded with the Medal of Anania Shirakatsi, by the decree of the President of the Republic of Armenia.

For many years, G. Karapetyan was a co-head of the scientific seminar “Functional spaces and their applications in mathematical physics” first in YSU, and then in RAU. He authored or co-authored 8 textbooks and manuals and more than 60 research articles. His research was mainly devoted to elliptic and hypoelliptic differential equations, the study of the properties of functions in different multianisotropic spaces, integral representations and embedding theorems for functions in multianisotropic spaces.

The limited size of this text makes possible only a brief description of some, but not all, of the studies that G.Karapetyan has conducted, often with his students, which resulted in significant advances and new deep results. His main research activity encompasses the following:

- Convergence of the Galerkin approximations to the solution of the Dirichlet problem for different classes of hypoelliptic equations;
- Liouville and Phragmén-Lindelöf type theorems for general regular equations;
- Behavior of the solutions of a certain class of hypoelliptic and μ-elliptic equations in bounded and unbounded domains;
- Estimates for the solutions of quasi-elliptic equations in Hölder spaces;
- Schauder-type estimates for the solutions of semi-elliptic equations with variable coefficients;
- Study of the solutions of hypoelliptic and degenerate equations in the half-space;
- Study of boundary value problems for regular quasilinear equations in unbounded domains;
- The proof of stabilization of the solutions for regular and hypoelliptic equations of different classes;
- Study of the solutions of nondegenerate equations depending on a parameter;
- Study of eigenvalues of self-adjoint semi-elliptic operators;
- The proof of multiplicative inequalities of Gagliardo-Nirenberg type in multianisotropic Sobolev spaces;
- Approximations of the solutions of semi-elliptic equations in the half-space;
- Computation of the index of semi-elliptic operators;
- Proof of the Noetherianness of semi-elliptic and nondegenerate operators;
- Application of the small parameter method for approximation of the eigenvalues of regular hypoelliptic operators;
- Integral representations of functions in different multianisotropic spaces and embedding theorems for them.

Garnik Karapetyan was a highly qualified professional in his field as well as a talented leader. He was distinguished by his personal quality of paying particular attention and respect to people and their problems. He was impeccably honest and principled in his work, tactful in the relationships with his family members, relatives, friends, colleagues and students. G.Karapetyan was tremendously charismatic; he enjoyed exceptional trust and reverence from students. Not only did he demonstrate genuine interest in everything new in science and life in general, but he also triggered the interest of his students and those surrounding him.

G.Karapetyan was not only well-known among scientists and academics in particular fields; the general public knew him thanks to the educational TV program “Open Lesson” broadcasted by the Public Television of Armenia.

It is particularly sorrowful for the friends and colleagues of G.Karapetyan that he left this world at the time of his creative ascent. During the last 5-6 years he actively conducted fruitful research on the integral representations of functions in multianisotropic Sobolev spaces and embedding theorems for such spaces. His recent work was is the field of the studies carried out by many distinguished researchers, like S.Sobolev, L.Schwartz, S.Nikolskii, S.Agmon, O.Besov, H.Triebel, E.Gagliardo, P.Lizorkin, I.Slobodetsky, I.Nechas, V.Burenkov, S.Uspenskii, V.Solonnikov, G.Demidenko, A.Kufner, and many others, who studied isotropic or anisotropic Sobolev spaces.

In recent years, G.Karapetyan obtained integral representations for functions in multianisotropic Sobolev spaces generated by regular Newton polyhedra, and, basing on these representations, he proved embedding theorems for such spaces. Some of the obtained results are not yet published. The further plans of G.Karapetyan included the description of the traces of functions in multianisotropic spaces on hypersurfaces of different dimensions, theorems on the continuation of functions outside their domain of definition, etc. We hope that this work will be continued by his disciples.

The family, relatives, friends, colleagues and students of Garnik Karapetyan as well as the Armenian mathematics suffered an irretrievable loss due to his untimely death.

Cherished memory of Garnik Karapetyan will forever remain in our hearts.

Adamyan G.V., Aramyan R.H., Avetisyan P.S., Babayan A.H., Baghdasaryan A.G., Berberyan S.L., Darbinyan A.A., Darbinyan A.R., Gevorgyan G.G., Ghazaryan E.M., Ghazaryan H.G., Harutyunyan K.V., Harutyunyan T.N., Margaryan V.N., Petrosyan H.A., Sandoyan E.M., Sargsyan G.Z., Tonoyan R.N., Tumanyan A.G.

**Prof. Stasys Rutkauskas
1951/06/06 – 2018/10/29**

Professor Dr. Stasys Rutkauskas was born on June 6, 1951, in Nečionių village, Širvintai region of Lithuania. He graduated from the secondary school in 1968 in the city of Širvintai. In 1973 he finished studies at Faculty of Mathematics of Vilnius University and acquired a diploma as mathematician. In 1974 -1977 he was a PhD student at the Institute of Mathematics and Informatics in Vilnius, and in 1978 he defended his PhD theses in Minsk University and received the doctoral degree in mathematics. From 1977 to 1980 he worked as a junior research fellow and starting from 1980 – as a senior researchfellow at the Institute of Mathematics and Informatics. In 1995-2005 he was the Scientific Secretary of the Institute, and in 2005, after defending the Habilitation Theses and getting the Doctor of Science degree, he became a vice-director for research of the Institute of Mathematics and Informatics (nowadays Vilnius University Institute of Data Science and Digital Technologies).

Prof. S. Rutkauskas has published more that 50 scientific papers, he presented many invited talks at international scientific conferences. The topics of his research were boundary value problems for degenerate elliptic systems, asymptotic behavior of their solutions near singular points. Working on these problems, Prof. S. Rutkauskas actively collaborated with mathematicians in other countries, many time he delivered special lecture courses at foreign universities.

Prof. S. Rutkauskas was a member of the editorial boards of several international mathematical journals, such as “Lithuanian Mathematical Journal’’, “Mathematical Modelling and Analysis”. Prof. S. Rutkauskas gave lectures to bachelor and master students in Vilnius University (VU), Vilnius Pedagogical University (VPU) and Vilnius Gediminas Technical University (VGTU). His textbook for PhD students “Asymptotic Methods for Ordinary Differential Equations” is very useful for young mathematicians. Prof. S. Rutkauskas was the leader of the scientific project “Informatics and Mathematics doctoral studies development (InMaDra)” supported by European structural funds.

He loved his family very much. He had two sons and three granddaughters. Prof. S. Rutkauskas was warm, loved by all, a modest man who left unfinished both works and ideas.

Gintautas Dzemyda

Prof. Dr. Habil.

Full member of the Lithuanian Academy of Sciences

Director of the Institute of Mathematics and Informatics

Head of the Systems Analysis Department